A non-linear singularity-preserving solution to the least-squares seismic imaging problem with sparseness and continuity constraints is proposed. The applied formalism explores curvelets as a directional frame that, by their sparsity on the image, and their invariance under the imaging operators, allows for a stable recovery of the amplitudes. Our method is based on the estimation of the normal operator in the form of an ’eigenvalue’ decomposition with curvelets as the ’eigenvectors’. Subsequently, we propose an inversion method that derives from estimation of the normal operator and is formulated as a convex optimization problem. Sparsity in the curvelet domain as well as continuity along the reflectors in the image domain are promoted as part of this optimization. Our method is tested with a reverse-time ’wave-equation’ migration code simulating the acoustic wave equation.

A method that can be used to detect damage in structures is developed. This method is based on the convexity of the geometric interpretation of the eigenvalue problem for undamped positive definite systems. The damage detection scheme establishes various damage scenarios which are used as failure sets. These scenarios are then compared to the structure's actual response by measuring the natural frequencies of the structure and using a Euclideian norm. Mathematical models were developed for application of the method on a cantilever beam. Damage occurring at a single location or in multiple locations was estalished and studied. Experimental verification was performed on serval prismatic beams in which the method provided adequate results. The exact location and extent of damage for several cases was predicted. When the method failed the prediction was very close to the actual condition in the structure. This method is easy to use and does not require a rigorous amount of instrumentation for obtaining the experimental data required in the detection scheme. / Ph. D.

The central theme of this thesis deals with problems related to the question, “Can one hear the shape of a drum?” first posed formally by Mark Kac in 1966. More precisely, can one determine the shape of a membrane with fixed boundary from the spectrum of the associated differential operator? For this paper, Kac received both the Lester Ford Award and the Chauvant Prize of the Mathematical Association of America. This problem has received a great deal of attention in the past forty years and has led to similar questions in completely different contexts such as “Can one hear the shape of a graph associated with the Schrödinger operator?”, “Can you hear the shape of your throat?”, “Can you feel the shape of a manifold with Brownian motion?”, “Can one hear the crack in a beam?”, “Can one hear into the sun?”, etc. Each of these topics deals with inverse eigenvalue problems or related inverse problems. For inverse problems in general, the problem may or may not have a solution, the solution may not be unique, and the solution does not necessarily depend continuously on perturbation of the data. For example, in the case of the drum, it has been shown that the answer to Kac’s question in general is “no.” However, if we restrict the class of drums, then the answer can be yes. This is typical of inverse problems when a priori information and restriction of the class of admissible solutions and/or data are used to make the problem well-posed. This thesis provides an analysis of shapes for which the answer to Kac's question is positive and a variety of interesting questions on this problem and its variants, including cases that remain open. This thesis also provides a synopsis and perspectives of other types of “can one hear” problems mentioned above. Another part of this thesis deals with aspects of direct problems related to musical instruments.

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A stability analysis and departure prediction method has been developed and coded in a MATLAB®-based software package called the Stability And Departure Analysis Tool using Eigenvalue Sensitivity (SADATES). Using eigenvalue and eigenvector analysis, SADATES is capable of performing a full-envelope stability analysis, returning both quantitative and qualitative data regarding the stability of the airplane at a static reference condition. SADATES not only supplies the analyst with information describing where and when an aircraft is likely to depart, but also information about the departure characteristics, enabling the analyst to design for better departure resistance. While the eigenvalue and eigenvector approach is straightforward, it is a broader approach than many traditional stability parameters, yielding more accurate and reliable results than traditional methods. SADATES also analyzes the aircraft dynamics from a standpoint of eigenvalue sensitivity. Using this feature, the analyst may directly study the impact of data uncertainty and non-zero angular rates on the nominal stability of the aircraft. Of particular interest are the effects of the dynamic damping derivatives, as these derivatives are particularly difficult to estimate. In addition, the effect of an unsteady reference condition may be examined by studying the sensitivity of the eigenvalues to changes in angular rates, thereby using a static approach to give answers to a dynamic problem. Given the development of eigenvalue sensitivity data, the analyst is able to determine the margin of error on nominal aircraft stability. The utility of the SADATES package is tested using aerodynamic data of the McDonnell-Douglas F / A-18C Hornet. Bare airframe, controls fixed stability is analyzed, and its sensitivity to data uncertainty and to non-zero angular rates is examined. The Hornet's bare airframe stability characteristics are then compared to those using an active feedback control system to drive an automatic leading and trailing edge flap schedule, demonstrating the accuracy and versatility of the program. / Master of Science

In this thesis, we applied local analysis tools (eigenvalue and eigenvalue elasticity analysis, global function elasticity/sensitivity analysis), and global analysis tools (deriving the location and stability of fixed points) to both aggregate and individual-level dynamic models of infectious diseases. We sought to use these methods to gain insight into the models and to evaluate the use of these methods to study their short-term and long-term dynamics and the influences of arameters on the models.<p> We found that eigenvalues are effective for understanding short-term behaviours of a nonlinear system, but less effective in providing insights of the long-term impacts of a parameter change on its behaviours. In term of disease control, local changes of behaviours, yielded from the changes of parameters based on eigenvalue elasticity, are able to alter behaviours in a short-term, especially in the period of a disease outbreak. While eigenvalue elasticity analysis can be helpful for understanding the impact of parameter changes for simple aggregate models, such analyses prove unwieldy and complicated, particularly for models with large number of state variables; and easily fall prey to eigenvalue multiplicity problems for large individual-based models, and istracting artifacts associated with small denominators. In response to these concerns, we introduced other local methods (global function elasticity/sensitivity analyses) that capture many of the advantages of eigenvalue elasticity methods with much greater simplicity. Unfortunately, parameter changes guided by these local analysis techniques are often insufficient to alter behaviours in the longer-term, such as when system behaviours approach stable endemic equilibria. By contrast, the global analytic tools, such as fixed point location and stability analysis, are effective for providing insights into the global behaviours of disease spread in the long-term, as well as their dependence on parameters. Using all of the above analysis as a toolset, we gained some possible insights into combination of local and global approaches. Choice of applying local or global analysis tools to infectious disease models is dependent on the specific target of policy makers as well as model type.

In this thesis, we applied local analysis tools (eigenvalue and eigenvalue elasticity analysis, global function elasticity/sensitivity analysis), and global analysis tools (deriving the location and stability of fixed points) to both aggregate and individual-level dynamic models of infectious diseases. We sought to use these methods to gain insight into the models and to evaluate the use of these methods to study their short-term and long-term dynamics and the influences of arameters on the models.<p> We found that eigenvalues are effective for understanding short-term behaviours of a nonlinear system, but less effective in providing insights of the long-term impacts of a parameter change on its behaviours. In term of disease control, local changes of behaviours, yielded from the changes of parameters based on eigenvalue elasticity, are able to alter behaviours in a short-term, especially in the period of a disease outbreak. While eigenvalue elasticity analysis can be helpful for understanding the impact of parameter changes for simple aggregate models, such analyses prove unwieldy and complicated, particularly for models with large number of state variables; and easily fall prey to eigenvalue multiplicity problems for large individual-based models, and istracting artifacts associated with small denominators. In response to these concerns, we introduced other local methods (global function elasticity/sensitivity analyses) that capture many of the advantages of eigenvalue elasticity methods with much greater simplicity. Unfortunately, parameter changes guided by these local analysis techniques are often insufficient to alter behaviours in the longer-term, such as when system behaviours approach stable endemic equilibria. By contrast, the global analytic tools, such as fixed point location and stability analysis, are effective for providing insights into the global behaviours of disease spread in the long-term, as well as their dependence on parameters. Using all of the above analysis as a toolset, we gained some possible insights into combination of local and global approaches. Choice of applying local or global analysis tools to infectious disease models is dependent on the specific target of policy makers as well as model type.

Using the idea that a quantum mechanical system drops to its ground state as its temperature goes to absolute zero several operators are devised to enable the approximation of the lowest order energy eigenstate of a given symmetry; as well as an approximation to the energy eigenvalue of the same order.

The linear stability of the Giesekus and linear Phan-Thien Tanner (PTT) fluid models is investigated for a number of planar Poiseuille flows in single, double and triple layered configurations. The Giesekus and PTT models involve parameters that can be used to fit shear and extensional data, thus making them suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem which is then solved using the QZ-algorithm. The eigenspectra are shown to comprise of continuous parts and discrete parts. The theoretical and numerical results are validated for the Oldroyd-B model, which is a simplified case of the Giesekus and PTT models, by comparing with results in the literature. The continuous and discrete parts of the eigenspectra are determined using a purely numerical scheme to solve the discretized eigenvalue problem. The continuous spectra are then more accurately determined using a semi-analytical scheme which uses an analytical solution of the Orr-Sommerfeld equation alongside a numerical solution for the base flow. A comprehensive survey of the effect of each shear thinning and extensional fluid parameter is undertaken and an instability is found for particular parameter values for the Giesekus fluid. A preliminary investigation of this instability is undertaken whereby the unstable discrete eigenvalue is investigated using an Orthonormal Runge-Kutta scheme within a shooting method which uses the results from the Chebyshev-QZ scheme as a starting point. The linear PTT fluid is found to be stable to infinitesimal disturbances within the range of shear-thinning and extensional parameters considered. The computational e ciency and accuracy of the numerical methods are also investigated.

O objetivo desta dissertação é estudar o problema de autovalor de Dirichlet para variedades riemannianas completas. Mais precisamente, pretendemos estudar uma cota por baixo para o -ésimo autovalor de um domínio limitado em uma variedade riemanniana completa. Tal cota é obtida fazendo-se uso de uma fórmula de recorrência de Cheng e Yang e um teorema de Nash. Ademais, pretendemos estudar uma desigualdade universal para os autovalores no espaço hiperbólico. / The goal of this dissertation is to study the Dirichlet eigenvalue problem for a complete riemannian manifold. More accurately, we intend to investigate a lower-bound for the -ℎ eigenvalue on a bounded domain in a complete riemannian manifold. Such a bound is obtained by making use of a recursion formula of Cheng and Yang and Nash’s Theorem. Furthermore, we study a universal inequality for eigenvalues of the Dirichlet eigenvalue problem on a bounded domain in a hyperbolic space (−1).